THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

388 CASIMIR EFFECT AND VACUUM ENERGY
3k/(8πGρcR 3 ) ∝ R, in the early Universe where R is small one has |ρ−ρc|/ρc ≪
1. At t = 1 s after the Big Bang this was about 10 −16 .
What is the reason for such fine tuning? The answer can be provided by the
inflationary scenario in which the curvature term is exponentially suppressed,
since the exponential inflation of the Universe simply irons out curved space to
make it extraordinarily flat. The analogy with quantum liquids suggests another
solution of the flatness problem.
According to the ‘cosmological principle’ the Universe must be homogeneous
and isotropic. This is strongly confirmed by the observed isotropy of cosmic
background radiation. Within general relativity the Robertson–Walker metric
describes the spatially homogeneous and isotropic distribution of matter and
thus satisfies the cosmological principle. The metric field is not homogeneous,
but in general relativity the property of the homogeneity must be determined in
a covariant way, i.e. it should not depend on coordinate transformation. In the
Robertson–Walker Universe the covariant quantity – the curvature – is constant,
and thus this Universe is homogeneous.
However, if general relativity is an effective theory, the invariance under the
coordinate transformations exists at low energy only. At higher energy, the contravariant
metric field g µν itself becomes the physical quantity: an external observer
belonging to the trans-Planckian world can distinguish between different
metrics even if they are equivalent for the inner low-energy observer. According
to eqn (29.40) the metric field is not homogeneous unless k =0.Ifk = 0
the ‘Planck’ observer views the Robertson–Walker metric as space dependent.
Moreover, the r 2 -dependence of the contravariant metric element g rr ∝ 1 − kr 2
implies a huge deformation of the ‘Planck liquid’, which is strongly prohibited.
That is why, according to the trans-Planckian physics, the Universe must be flat
at all times, i.e. k = 0 and ρΛ + ρM = ρc. This means that the cosmological
principle of homogeneity of the Universe can well be an emergent phenomenon
reflecting the Planckian physics.
29.4.7 What is the energy of false vacuum?
It is commonly believed that the vacuum of the Universe underwent one or several
broken-symmetry phase transitions. Each of the transitions is accompanied by
a substantial change in the vacuum energy. Moreover, there can be false vacua
separated from the true vacuum by a large energy barrier. Why is the true
vacuum so distinguished from the others that it has exactly zero energy, while the
energies of all other false vacua are enormously large? Where does the principal
difference between vacua come from?
The quantum liquid answer to this question is paradoxical (see Sec. 29.2.2):
in the absence of external forces all the vacua including the false ones have
zero energy density and thus zero cosmological constant. There is no paradox,
however, because the positive energy difference between the false and the true
vacuum is obtained at fixed chemical potential µ. Let us suppose that the liquid
is in the false vacuum state. Then its vacuum energy density ˜ɛ = ɛ − µn ≡
√ −gρΛ = 0. After the transition from the false vacuum to the true one has